Many-body dynamics of association in quantum gases

Many-body dynamics of association in quantum gases

E. Pazy, I. Tikhonenkov, Y. B. Band, M. Fleischhauer, and A. Vardi

E. Pazy, I. Tikhonenkov, Y. B. Band, M. Fleischhauer, and A. Vardi

OutlineOutline Fermion association model - single molecular mode, slow

atoms. Equivalence with the Dicke problem (Fermion atoms)

and parametric downconversion (Boson atoms). Undepleted pump dynamics and SU(2)/SU(1,1) coherent

states. Pump depletion and Fermion-Boson mapping. Adiabatic sweep dynamics - breakdown of Landau-Zener

and appearance of power-laws instead of exponentials.

Fermion association model - single molecular mode, slow atoms.

Equivalence with the Dicke problem (Fermion atoms) and parametric downconversion (Boson atoms).

Undepleted pump dynamics and SU(2)/SU(1,1) coherent states.

Pump depletion and Fermion-Boson mapping. Adiabatic sweep dynamics - breakdown of Landau-Zener

and appearance of power-laws instead of exponentials.

Fermion AssociationFermion Association

Expand in eigenmodes of free Hamiltonians (e.g. plane waves for a uniform gas):

Any pair of atoms with opposite spins can associate !

Single boson mode approximationSingle boson mode approximationCooper instability for zero center of mass motion:

k

-k

k1=k+q

-k1=-k-q

k’

-k’

|q|

Single boson mode approximationSingle boson mode approximation

|q|

K/2+k

K/2+k+q

K/2-k-q

K/2-k

K/2+k+q’

K/2-k-q’

Zero relative motion does not coincide withthe discontinuity in the momentum distribution.

Non-zero center of mass motion:

Instability washes out !

Single boson mode approximationSingle boson mode approximation

Starting from a quantum degenerate Fermi gas, atoms are paired with opposite momentum.

Experimentally, a molecular BEC is formed at low T.

Starting from a quantum degenerate Fermi gas, atoms are paired with opposite momentum.

Experimentally, a molecular BEC is formed at low T.

Pseudospin representationPseudospin representationAnderson, Phys. Rev. 112, 1900 (1958).Barankov and Levitov, Phys. Rev. Lett. 93, 130403 (2004).

Fermion-pair operators generate SU(2)

Tavis-Cummingsmodel:

Phys. Rev. 170, 379 (1968).

Slow atomic dispersionSlow atomic dispersion

For g >> Ef (slow atomic motion compared with atom-molecule conversion, i.e Raman-Nath approximation), Tavis-Cummings reduces to the Dicke model:

For g >> Ef (slow atomic motion compared with atom-molecule conversion, i.e Raman-Nath approximation), Tavis-Cummings reduces to the Dicke model:

Comparison with boson pairsComparison with boson pairs

Boson-pair operators generate SU(1,1)

Comparison of degenerate modelsComparison of degenerate models

Two-mode bosons - SU(1,1) - Parametric downconversion

Degenerate fermions - SU(2) - Dicke model

Starting from the atomic vacuum (i.e. a molecular BEC) and neglecting molecular depletion, fermion atoms will be in SU(2) coherent states whereas boson atoms will form SU(1,1) coherent states.

Fermions: SU(2) Coherent StatesFermions: SU(2) Coherent States

<Jx>/M<Jy>/M

<J z>

/M

Bosons: SU(1,1) Coherent StatesBosons: SU(1,1) Coherent States

<Kx>/M<Ky>/M

<K

z>/M

Number statistics - on resonance, starting from the atomic vacuumNumber statistics - on resonance, starting from the atomic vacuum

Fermions Bosons

Unstable molecular modeAmplified quantum noise

Unstable molecular modeAmplified quantum noise

Stable molecular modeBounded fluctuations

Stable molecular modeBounded fluctuations

Bose stimulationBose stimulation

‘Thermal gas’

Bose stimulationBose stimulation

‘Molecular BEC’

Include molecular depletionInclude molecular depletionFermions Bosons

Fermion-Boson mappingFermion-Boson mapping Boson states with n atom-pairs map exactly to fermion

states with n hole-pairs. Boson dissociation dynamics (starting with the atomic

vacuum) is identical to fermion association dynamics (starting with the molecular vacuum) and vice versa.

Boson states with n atom-pairs map exactly to fermion states with n hole-pairs.

Boson dissociation dynamics (starting with the atomic vacuum) is identical to fermion association dynamics (starting with the molecular vacuum) and vice versa.

Fermion association Boson dissociation

Atom-molecule adiabatic sweepsAtom-molecule adiabatic sweeps

atoms

atoms

molecules

molecules

3 MF eigenvalues2 elliptic, 1 hyperbolic

2 stable eigenstates

2 stable eigenstates

curves cross !

10 pairs eigenvalues - EF=0 Reduced single-pair (mean-field) picture

Many-body adiabatic passageMany-body adiabatic passage

Classical (mean-field) limitClassical (mean-field) limit

Classical eigenstatesClassical eigenstates

or

- Two elliptic fixed points

- Two elliptic and one hyperbolic FP

Classical phase-space structure

Classical phase-space structure

Adiabatic MF eigenvalues

Nonadiabaticity and actionNonadiabaticity and actionTransform to action-angle variables:

The square of the remnant atomic fraction is proportional to the action accumulated during the sweep.

- rate of change of fixed points

- characteristic precession frequency

Adiabatic eigenstates should be varied slowly with respectto characteristic frequencies around them !

Why 2 ?Why 2 ?

u

v

w

Bloch sphere conservation law:

Why 2 ?Why 2 ?

But we have a different Casimir :

Landau-ZenerLandau-Zener

Re(

Im(

The integrand has no singularities on the real axis

is exponentially smallis exponentially small

Breakdown of Landau-ZenerBreakdown of Landau-Zener

Landau Zener : In our nonlinear system, we have homoclinic orbits

starting and ending at hyperbolic fixed points.

The period of homoclinic orbits diverges - i.e. the characteristic frequency vanishes near these points

Consequently there are real singularities.

Landau Zener : In our nonlinear system, we have homoclinic orbits

starting and ending at hyperbolic fixed points.

The period of homoclinic orbits diverges - i.e. the characteristic frequency vanishes near these points

Consequently there are real singularities.

Power-law dependence - analysisPower-law dependence - analysis

- Independent of

Power-law dependence - analysisPower-law dependence - analysis

When 1-w0(ti ) >> 1/N (neglecting noise):

When 1-w0(ti ) << 1/N (including noise term):

Power-law dependence - numericsPower-law dependence - numerics

Comparison with experimentComparison with experiment

ConclusionsConclusions

Short-time collective dynamics is significantly different for fermions and bosons (Pauli blocking vs. Bose stimulation).

But, there is a mapping between fermion association and boson dissociation and vice versa.

And, nonlinear effects modify the dependence of conversion efficiency on sweep rates, rendering the Landau-Zener picture inapplicable and leading to power-laws instead of exponents.

Short-time collective dynamics is significantly different for fermions and bosons (Pauli blocking vs. Bose stimulation).

But, there is a mapping between fermion association and boson dissociation and vice versa.

And, nonlinear effects modify the dependence of conversion efficiency on sweep rates, rendering the Landau-Zener picture inapplicable and leading to power-laws instead of exponents.

Students/Postdocs Wanted !Students/Postdocs Wanted !

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